Two long wires lie in an xy plane, and each carries a...

Two long wires lie in an xy plane, and each carries a current in the positive direction of the $x$ axis. Wire 1 is at $y=10.0 \mathrm{cm}$ and carries 6.00 $\mathrm{A}$ ; wire 2 is at $y=5.00 \mathrm{cm}$ and carries 10.0 $\mathrm{A}$ . (a) In unit- vector notation, what is the net magnetic field $\vec{B}$ at the origin? (b) At what value of $y$ does $\vec{B}=0 ?$ (c) If the current in wire 1 is reversed, at what value of $y$ does $\vec{B}=0 ?$

Two long wires lie in an xy plane, and each carries a current in the positive direction of the $x$ axis. Wire 1 is at $y=10.0 \mathrm{cm}$ and carries 6.00 $\mathrm{A}$ ; wire 2 is at $y=5.00 \mathrm{cm}$ and carries 10.0 $\mathrm{A}$ . (a) In unit-
vector notation, what is the net magnetic field $\vec{B}$ at the origin? (b)
At what value of $y$ does $\vec{B}=0 ?$ (c) If the current in wire 1 is reversed, at what value of $y$ does $\vec{B}=0 ?$

Key Concepts

Unit Vector Notation

Unit vector notation expresses vectors in terms of their magnitude and direction. In the context of magnetic fields, it is used to clearly represent the magnitude and directional components (e.g., in the i, j, k directions) of the field.

Effect of Current Direction on Magnetic Field Orientation

Reversing the direction of the current in a wire reverses the direction of the magnetic field it produces according to the right-hand rule. This concept is crucial when analyzing how different current orientations can lead to constructive or destructive interference of magnetic fields.

Superposition Principle

This key concept states that the net magnetic field due to multiple sources is the vector sum of the fields produced by the individual sources. It allows one to compute the overall magnetic field by adding contributions from each wire independently.

Right-Hand Rule

This is a method used to determine the direction of the magnetic field around a current-carrying wire. By pointing the thumb of the right hand in the direction of the current, the curl of the fingers shows the direction in which the magnetic field encircles the wire.

Magnetic Field of a Current-Carrying Conductor

This concept involves determining the magnetic field produced by an infinitely long, straight current-carrying wire. The magnetic field at a distance r is given by the formula B = (??I)/(2?r), where ?? is the permeability of free space and I is the current. This is a cornerstone result derived from the Biot–Savart law or Ampere’s law.

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